Numbers 1 to 32 are placed along the circumference of a circle without repeating any number and still the sum of any two adjacent numbers in this circle is a perfect square!

## Sum of Consecutive Cubes (Visual Proof)

The sum of the first *n* cubes is the square of the *n*th triangular number:

1^{3} + 2^{3} + 3^{3} + 4^{3} + 5^{3} + . . . + *n*^{3} = (1 + 2 + 3 + 4 + 5 + . . . + *n*)^{2}.

## Inverse Powers of Phi

Summation of Alternating Inverse Powers of Phi…

## Repunit Primes

Write the digit “1” exactly 317 times, and you get a palindromic prime number. Moreover, 317 itself is a prime number!

## Visual Proof (sum of cubes)

The sum of the sequence of the first *n* cubes equals [*n*(*n*+1)/2]² as shown below:

1³+2³+3³+…+*n*³ = (1+2+3+…+*n*)² = [*n*(*n*+1)/2]²

## Infinite Pythagorean Triplets

Consider the following simple progression of whole and fractional numbers (with odd denominators):

1 1/3, 2 2/5, 3 3/7, 4 4/9, 5 5/11, 6 6/13, 7 7/15, 8 8/17, 9 9/19, …

Any term of this progression can produce a Pythagorean triplet, for instance:

4 4/9 = 40/9; the numbers 40 and 9 are the sides of a right triangle, and the hypotenuse is one greater than the largest side (40 + 1 = 41).

## Cauchy Product

## Fibonacci Spiral Jigsaw Puzzle

Each piece of this puzzle is similar (the same shape at a different size). The placement of the pieces is based on the golden angle (≈137.5º), and results in a pattern frequently found in nature (phyllotaxis), for instance on sunflowers. The puzzle features 8 spirals in one direction, and 13 in the other. You can build your own Fibonacci spiral puzzle by following John Edmark’s tutorial.

## Circles and Golden Ratio

The last digit of the numbers in the Fibonacci Sequence are cyclic, they form a pattern that repeats after every 60th number: 0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1.

## Surprising Limit

Amazingly, this sequence of fractions converges to 0.70710678118…, or to be precise, to √2/2. The sequence is related to the Prouhet-Thue-Morse sequence.